# An overview of Koopman-based control: From error bounds to closed-loop guarantees

arXiv: https://arxiv.org/abs/2509.02839 · HTML: https://mlsanity.com/paper/2509.02839/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2025.

## Built on by (most cited first, 10 of 10)

- [InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics](https://mlsanity.com/paper/2510.25943/) (2025)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [An Output Feedback Q-learning Algorithm for Optimal Control of Nonlinear Systems with Koopman Linear Embedding](https://mlsanity.com/paper/2603.29858/) (2026)
- [Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors](https://mlsanity.com/paper/2604.01459/) (2026)
- [On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations](https://mlsanity.com/paper/2604.03086/) (2026)
- [Dynamic Neural Koopman Distillation for Real-Time Robot Control Using Diffusion Models](https://mlsanity.com/paper/2605.24924/) (2026)
- [Learning to control switching nonlinear systems with Koopman operator regression](https://mlsanity.com/paper/2607.11344/) (2026)
- [Beyond Linear Dynamics: Neural Bilinear Dynamical Models for Time Series Forecasting](https://mlsanity.com/paper/2608.04471/) (2026)
- [Demystifying Linear Operator Learning for Control Systems](https://mlsanity.com/paper/2609.19428/) (2026)

## Abstract (arXiv)

Controlling nonlinear dynamical systems remains a central challenge in a wide range of applications, particularly when accurate first-principle models are unavailable. Data-driven approaches offer a promising alternative by designing controllers directly from observed trajectories. A wide range of data-driven methods relies on the Koopman-operator framework that enables linear representations of nonlinear dynamics via lifting into higher-dimensional observable spaces. Finite-dimensional approximations, such as extended dynamic mode decomposition (EDMD) and its controlled variants, make prediction and feedback control tractable but introduce approximation errors that must be accounted for to provide rigorous closed-loop guarantees. This survey provides a systematic overview of Koopman-based control, emphasizing the connection between data-driven surrogate models, approximation errors, controller design, and closed-loop guarantees. We review theoretical foundations, error bounds, and both linear and bilinear EDMD-based control schemes, highlighting robust strategies that ensure stability and performance. Finally, we discuss open challenges and future directions at the interface of operator theory, approximation theory, and nonlinear control.